Preface
Acknowledgments
Contents
About the Authors
1 A Review of Commutative Harmonic Analysis
1.1 The Haar Measure
1.1.1 Locally Compact Abelian Groups
1.1.2 Existence and Properties of the Haar Measure
1.2 The Dual Group and the Fourier Transform
1.2.1 Characters and the Algebra L1(G)
1.2.2 Topology on the Dual Group
1.2.3 Examples and Basic Facts
1.3 The Bochner–Weil–Raikov and Peter–Weyl Theorems
1.3.1 An Abstract Theorem
1.3.2 Applications to Harmonic Analysis
1.4 The Inversion and Plancherel Theorems
1.4.1 The Inversion Theorem
1.4.2 The Plancherel Theorem
1.4.3 A Description of the Wiener Algebra
1.4.4 A Basic Hilbert-Type Inequality
1.5 Pontryagin's Duality Theorem and Applications
1.5.1 The Pontryagin Theorem
1.5.2 Topological Applications of Pontryagin's Theorem
1.6 The Uncertainty Principle
1.6.1 The Uncertainty Principle on the Real Line
1.6.2 The Uncertainty Principle on Finite Groups
1.7 Exercises
References
2 Ergodic Theory and Kronecker's Theorems
2.1 Elements of Ergodic Theory
2.1.1 Basic Notions in Ergodic Theory
2.1.2 Ergodic Theorems
2.2 The Kronecker Theorem
2.2.1 Definitions
2.2.2 Statement and Proof of the Main Theorem
2.2.3 A Useful Formulation of Kronecker's Theorem
2.3 Distribution Problems
2.3.1 Distribution in mathbbTd
2.3.2 Powers of an Algebraic Number
2.4 Towards Infinite Dimension
2.5 Exercises
References
3 Diophantine Approximation
3.1 One-Dimensional Diophantine Approximation
3.1.1 Historical Survey
3.1.2 How to Find the Best Approximations?
3.1.3 Classification of Numbers
3.1.4 First Arithmetical Results
3.2 The Gauss Ergodic System
3.3 Back to Transcendence
3.3.1 Metric Results
3.3.2 Simultaneous Diophantine Approximations
3.4 Exercises
References
4 General Properties of Dirichlet Series
4.1 Introduction
4.2 Convergence Abscissas
4.2.1 The Bohr–Cahen Formulas
4.2.2 The Perron–Landau Formula
4.2.3 The Holomorphy Abscissa
4.2.4 A Class of Examples
4.2.5 A More Intricate Dirichlet Series
4.2.6 Automatic Dirichlet Series
4.3 Products of Dirichlet Series
4.4 Bohr's Abscissa via Kronecker's Theorem
4.4.1 Bohr's Point of View
4.4.2 The Bohr Inequalities
4.4.3 A Wiener Lemma for Dirichlet Series
4.5 A Theorem of Bohr and Jessen on Zeta
4.6 Exercises
References
5 Probabilistic Methods for Dirichlet Series
5.1 Introduction
5.2 A Multidimensional Bernstein Inequality
5.3 Random Polynomials
5.3.1 Maximal Functions in Probability
5.3.2 The Sub-Gaussian Aspect of Rademacher-Type Variables
5.3.3 The Kahane Bound for Random Trigonometric Polynomials
5.3.4 Random Dirichlet Polynomials
5.4 The Proof of Bohnenblust–Hille's Theorem
5.4.1 An Elementary Version
5.4.2 A Sharp Version of the Bohnenblust–Hille Theorem
5.5 Exercises
References
6 Hardy Spaces of Dirichlet Series
6.1 Definition and First Properties
6.1.1 The Origin of the Spaces mathcalHinfty and mathcalH2
6.1.2 A Basic Property of mathcalHinfty
6.2 The Banach Space mathcalHinfty
6.2.1 The Banach Algebra Structure of mathcalHinfty
6.2.2 Behaviour of Partial Sums
6.2.3 Some Applications of the Control of Partial Sums
6.3 Additional Properties of mathcalHinfty
6.3.1 An Improved Montel Principle
6.3.2 Interpolating Sequences of mathcalHinfty
6.4 The Hilbert Space mathcalH2
6.4.1 Definition and Utility
6.4.2 The Embedding Theorem
6.4.3 Multipliers of mathcalH2
6.5 The Banach Spaces mathcalHp
6.5.1 A Basic Identity
6.5.2 Definition of Hardy-Dirichlet Spaces
6.5.3 A Detour Through Harmonic Analysis
6.5.4 Helson Forms
6.5.5 Harper's Breakthrough and Two Consequences
6.6 A Sharp Sidon Constant
6.6.1 Reformulation of Bohr's Question in Terms of mathcalHinfty
6.6.2 Symmetric Multilinear Forms
6.6.3 The Claimed Sharp Upper Bound
6.6.4 A Refined Inclusion
6.7 Exercises
References
7 Voronin-Type Theorems
7.1 Introduction
7.1.1 A Reminder About Zeta and L-Functions
7.1.2 Universality
7.2 Hilbertian Results
7.2.1 A Hilbertian Density Criterion
7.2.2 A Density Result in Bergman Spaces
7.3 Joint Universality of the Sequence (λN)
7.4 A Generalized and Uniform Carlson Formula
7.4.1 Estimates on the Gamma Function
7.4.2 The Carlson Formula
7.5 Joint Universality of the Singleton λ= (L(s, χj))
7.5.1 Notations and the Idea of Proof of Theorem 1.2
7.5.2 Details of Proof
7.6 Exercises
References
8 Composition Operators on the Space mathcalH2 of Dirichlet Series
8.1 Introduction
8.2 The Main Theorem
8.3 The Arithmetic Theorem
8.4 Twisting
8.4.1 Definition and Uniform Convergence
8.4.2 Mapping Properties
8.4.3 Twisting and Composition
8.4.4 Twisting and Probability
8.4.5 Twisting and Topology
8.5 Integral Representation and Embedding
8.5.1 Integral Representation
8.5.2 Embedding Results for mathcalH2
8.6 Proof of the Main Theorem
8.6.1 Proof of Necessity in Theorem 8.6.1
8.6.2 Proof of Sufficiency in Theorem 8.6.1
8.7 Compact Operators and Approximation Numbers
8.7.1 Definitions
8.7.2 First Properties of Approximation Numbers
8.7.3 The Multiplicative Inequalities of H. Weyl
8.8 A Lower Bound
8.9 Upper Bounds
8.10 The Case of mathcalHp-spaces
8.10.1 Reminder
8.10.2 Failure of Embedding for 0
8.10.3 Positive Results
8.11 A Few Updates and Remarks
8.12 Exercises
References
Index
